{"status": "success", "data": {"description_md": "For any positive integer $n\\ge 2$ let $d(n)$ denote the number of positive divisor of $n$ and $\\phi(n)$ denote the number of positive integers from $1$ to $n$ inclusive relatively prime with $n$. Let $\\nu_p(n)$ denote the largest value $k$ such that $p^k$ divides $n$. Given that $N$ is a positive integer for which $d(N)=2025$, let the minimum value of $\\phi(N)$ be $x$. Find\n\n$$\\sum_{p \\text{ is prime}}{\\nu_p(x)}$$", "description_html": "<p>For any positive integer <span class=\"katex--inline\">n\\ge 2</span> let <span class=\"katex--inline\">d(n)</span> denote the number of positive divisor of <span class=\"katex--inline\">n</span> and <span class=\"katex--inline\">\\phi(n)</span> denote the number of positive integers from <span class=\"katex--inline\">1</span> to <span class=\"katex--inline\">n</span> inclusive relatively prime with <span class=\"katex--inline\">n</span>. Let <span class=\"katex--inline\">\\nu_p(n)</span> denote the largest value <span class=\"katex--inline\">k</span> such that <span class=\"katex--inline\">p^k</span> divides <span class=\"katex--inline\">n</span>. Given that <span class=\"katex--inline\">N</span> is a positive integer for which <span class=\"katex--inline\">d(N)=2025</span>, let the minimum value of <span class=\"katex--inline\">\\phi(N)</span> be <span class=\"katex--inline\">x</span>. Find</p>&#10;<p><span class=\"katex--display\">\\sum_{p \\text{ is prime}}{\\nu_p(x)}</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "Grinding Aces Math Exam 2025 - Problem 5", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}